Slide 7- 1 Copyright © 2012 Pearson Education, Inc.

4.2 Solving Equations and Inequalities by Graphing ■ Solving Equations Graphically: The Intersect Method ■ Solving Equations Graphically: The Zero Method ■ Solving Inequalities Graphically ■ Applications

Slide 4- 3 Copyright © 2012 Pearson Education, Inc. Solving Equations Graphically: The Intersect Method Recall that to solve an equation or inequality means to find all the replacements for the variable that make the equation or inequality true. We have seen how to do this algebraically; we now use a graphical method to solve.

Slide 4- 4 Copyright © 2012 Pearson Education, Inc. Example Solve graphically: Solution Graph and g(x) = 2 on the same set of axes. The intersection appears to be (  4, 2), the solution is apparently  4. Check:

Slide 4- 5 Copyright © 2012 Pearson Education, Inc. Example Solve: Solution Algebraic Approach Check:

Slide 4- 6 Copyright © 2012 Pearson Education, Inc. Solution Graphical Approach We graph: It appears that the lines intersect at (6, 7). Check x = 6 in the equations and it checks. The solution is x = 6

Slide 4- 7 Copyright © 2012 Pearson Education, Inc. CAUTION! When using a hand-drawn graph to solve an equation, it is important to use graph paper and to work as neatly as possible. Use a straightedge when drawing lines and be sure to erase any mistakes.

Slide 4- 8 Copyright © 2012 Pearson Education, Inc. Example Solve using a graphing calculator. Solution We graph: It appears from the screen that the solution is

Slide 4- 9 Copyright © 2012 Pearson Education, Inc. Solving Equations Graphically: The Zero Method It can be challenging to determine a portion of the x, y- coordinate plane that contains the point of intersection. The Zero method make that determination easier because we are only interested in the point at which the graph crosses the x-axis.

Slide Copyright © 2012 Pearson Education, Inc. Example Solve graphically, using the Zero method: 3x – 2 = 5x – 7 Solution We first get 0 on one side of the equation. 3x – 2 = 5x – 7 –2x – 2 = –7 Subtracting 5x from both sides –2x + 5 = 0 Adding 7 to both sides Graph f(x) = –2x + 5, and find the x-intercept. The x-intercept of the graph appears to be (2.5, 0). We check 2.5 in the original equation and it checks.

Slide Copyright © 2012 Pearson Education, Inc. Example Solve graphically: 12 – 3x > 6x – 5. Solution We let y 1 = 12 – 3x and y 2 = 6x – 5, and graph y 1 and y 2. To the left of the point of intersection, y 1 > y 2. The solution set will be all x-values to the left of the point of intersection. ( , 1.889) Solving Inequalities Graphically

Slide Copyright © 2012 Pearson Education, Inc. Example An computer repair firm charges $35 plus $80 per hour for computer repairs. Formulate and graph a mathematical model for the cost. Then use the model to estimate the time required for the total cost to reach $ Familiarize. The problem describes a situation in which a firm charges $35 and then $80 for each hour of work. 1 hour: $ (1) = $115 2 hours: $ (2) = 195 Generalize a model if C(t) = the total cost for t hours of work. Applications

Slide Copyright © 2012 Pearson Education, Inc. continued 2. Translate. We reword and translate as follows: Rewording: the initial $80 per The total cost is charge plus hour Translating: c(t) = $35 + $80t where t  0 (since there can not be a negative number of hours) 3. Carry out. Estimate the solution by graphing C(t) = $ t and y = $325 and look for the point of intersection.

Slide Copyright © 2012 Pearson Education, Inc. continued Graphs: 4. Check. We evaluate: C(3.625) = (3.625) = = State. It takes hours for the total cost to reach $325.